Approximating the Sum of a Convergent Alternating Series

Approximate the sum \(S\) of the alternating series correct to within 0.0001. \[ \sum\limits_{k = 0}^\infty {\frac{{{{( - 1)}^k}}}{{(2k)!}} = 1 - \frac{1}{{2!}} + \frac{1}{{4!}} - \frac{1}{{6!}} + \frac{1}{{8!}} - ...} \]

Solution We demonstrated in Example 2 that this series converges by showing that it satisfies the conditions of the Alternating Series Test. The fifth term of the series, \(\dfrac{1}{8!}=\dfrac{1}{40,\!320}\approx 0.000025\), is the first term less than or equal to \(0.0001\). This term represents an upper estimate to the error when the sum \(S\) of the series is approximated by adding the first four terms. So, the sum \[ S \approx \sum\limits_{k = 0}^3 {\frac{{{{( - 1)}^k}}}{{(2k)!}} = 1 - \frac{1}{{2!}} + \frac{1}{{4!}} - \frac{1}{{6!}} = 1 - \frac{1}{2} + \frac{1}{{24}} - \frac{1}{{720}} = \frac{{389}}{{720}} \approx 0.5403} \]

is correct to within 0.0001.